32 found
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  1.  34
    On the cofinality of ultrapowers.Andreas Blass & Heike Mildenberger - 1999 - Journal of Symbolic Logic 64 (2):727-736.
    We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g. We also prove some related results for reduced powers with respect to filters other than ultrafilters.
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  2.  3
    Higher Miller forcing may collapse cardinals.Heike Mildenberger & Saharon Shelah - 2021 - Journal of Symbolic Logic 86 (4):1721-1744.
    We show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$. We show that under $\kappa ^{ \kappa $, club $\kappa $ -Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $. Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.
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  3.  7
    Covering the Baire space by families which are not finitely dominating.Heike Mildenberger, Saharon Shelah & Boaz Tsaban - 2006 - Annals of Pure and Applied Logic 140 (1):60-71.
    It is consistent that each union of many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter , the cofinality of the reduced ultrapower is greater than . The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.
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  4.  27
    On Milliken-Taylor Ultrafilters.Heike Mildenberger - 2011 - Notre Dame Journal of Formal Logic 52 (4):381-394.
    We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k -colored Milliken-Taylor ultrafilters have at least k +1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model.
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  5.  34
    Changing cardinal invariants of the reals without changing cardinals or the reals.Heike Mildenberger - 1998 - Journal of Symbolic Logic 63 (2):593-599.
    We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changing b and d from a regular κ to some regular δ < κ is a measurable of Mitchell order δ. There is an application to Cichon's diagram.
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  6.  31
    No Borel Connections for the Unsplitting Relations.Heike Mildenberger - 2002 - Mathematical Logic Quarterly 48 (4):517-521.
    We prove that there is no Borel connection for non-trivial pairs of unsplitting relations. This was conjectured in [3].
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  7.  44
    Specialising Aronszajn trees by countable approximations.Heike Mildenberger & Saharon Shelah - 2003 - Archive for Mathematical Logic 42 (7):627-647.
    We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.
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  8.  25
    Changing cardinal characteristics without changing ω-sequences or cofinalities.Heike Mildenberger & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 106 (1-3):207-261.
    We show: There are pairs of universes V1V2 and there is a notion of forcing PV1 such that the change mentioned in the title occurs when going from V1[G] to V2[G] for a P-generic filter G over V2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
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  9.  51
    Groupwise dense families.Heike Mildenberger - 2001 - Archive for Mathematical Logic 40 (2):93-112.
    We show that the Filter Dichotomy Principle implies that there are exactly four classes of ideals in the set of increasing functions from the natural numbers. We thus answer two open questions on consequences of ? < ?. We show that ? < ? implies that ? = ?, and that Filter Dichotomy together with ? < ? implies ? < ?. The technical means is the investigation of groupwise dense sets, ideals, filters and ultrafilters. With related techniques we prove (...)
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  10.  8
    More canonical forms and dense free subsets.Heike Mildenberger - 2004 - Annals of Pure and Applied Logic 125 (1-3):75-99.
    Assuming the existence of ω compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on ω. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. thm2.1Suppose the consistency of “ZFC+GCH + there are infinitely many compact cardinals”. Then the following is consistent: ZFC+GCH + and for every family 0 (...))
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  11. Notre Dame, Indiana May 20–May 23, 2009.Patricia Blanchette, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber - 2009 - Bulletin of Symbolic Logic 15 (4).
     
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  12.  6
    Uncountable structures are not classifiable up to bi-embeddability.Filippo Calderoni, Heike Mildenberger & Luca Motto Ros - 2019 - Journal of Mathematical Logic 20 (1):2050001.
    Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic164(12) (2013) 1454–1492], we show that whenever κ is a cardinal satisfying κ<κ=κ>ω, then the embeddability relation between κ-sized structures is strongly invariantly universal, and hence complete for (κ-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or (...)
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  13.  40
    There May be Infinitely Many Near-Coherence Classes under u < ∂.Heike Mildenberger - 2007 - Journal of Symbolic Logic 72 (4):1228 - 1238.
    We show that in the models of u < ∂ from [14] there are infinitely many near-coherence classes of ultrafilters, thus answering Banakh's and Blass' Question 30 of [3] negatively. By an unpublished result of Canjar, there are at least two classes in these models.
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  14.  4
    Mathias and silver forcing parametrized by density.Giorgio Laguzzi, Heike Mildenberger & Brendan Stuber-Rousselle - 2023 - Archive for Mathematical Logic 62 (7):965-990.
    We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse $$2^\omega $$ 2 ω to $$\omega $$ ω, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.
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  15.  3
    On splitting trees.Giorgio Laguzzi, Heike Mildenberger & Brendan Stuber-Rousselle - 2023 - Mathematical Logic Quarterly 69 (1):15-30.
    We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well‐known notions, such as Lebesgue measurablility, Baire‐ and Doughnut‐property and the Marczewski field. Moreover, we prove that any absolute amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Hein's and Spinas' conjecture that.
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  16.  6
    A version of $$kappa $$ κ -Miller forcing.Heike Mildenberger & Saharon Shelah - 2020 - Archive for Mathematical Logic 59 (7-8):879-892.
    We consider a version of \-Miller forcing on an uncountable cardinal \. We show that under \, \, and forcing with \\) collapses \ to \.
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  17.  11
    Borel on the Questions Versus Borel on the Answers.Heike Mildenberger - 1999 - Mathematical Logic Quarterly 45 (1):127-133.
    We consider morphisms between binary relations that are used in the theory of cardinal characteristics. In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions. In this work we investigate whether the second half, the function on the answers' side, has a similarly strong impact. The main question is: Does the (...)
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  18.  9
    Creatures on ω 1 and weak diamonds.Heike Mildenberger - 2009 - Journal of Symbolic Logic 74 (1):1-16.
    We specialise Aronszajn trees by an $\omega ^\omega $ -bounding forcing that adds reals. We work with creature forcings on uncountable spaces. As an application of these notions of forcing, we answer a question of Moore, Hrušák and Džamonja whether ◇(b) implies the existence of a Souslin tree in a negative way by showing that "◇∂ and every Aronszajn tree is special" is consistent relative to ZFC.
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  19.  5
    Exactly two and exactly three near-coherence classes.Heike Mildenberger - 2023 - Journal of Mathematical Logic 24 (1).
    We prove that for [Formula: see text] and [Formula: see text] there is a forcing extension with exactly n near-coherence classes of non-principal ultrafilters. We introduce localized versions of Matet forcing and we develop Ramsey spaces of names. The evaluation of some of the new forcings is based on a relative of Hindman’s theorem due to Blass 1987.
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  20.  11
    Finding generic filters by playing games.Heike Mildenberger - 2010 - Archive for Mathematical Logic 49 (1):91-118.
    We give some restrictions for the search for a model of the club principle with no Souslin trees. We show that ${\diamondsuit(2^\omega, [\omega]^\omega}$ , is almost constant on) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. This implies the analogous result for a double weakening of the club principle.
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  21.  4
    Increasing the groupwise density number by c.c.c. forcing.Heike Mildenberger & Saharon Shelah - 2007 - Annals of Pure and Applied Logic 149 (1-3):7-13.
  22.  13
    Many countable support iterations of proper forcings preserve Souslin trees.Heike Mildenberger & Saharon Shelah - 2014 - Annals of Pure and Applied Logic 165 (2):573-608.
    We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
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  23.  56
    Meeting infinitely many cells of a partition once.Heike Mildenberger & Otmar Spinas - 1998 - Archive for Mathematical Logic 37 (7):495-503.
    We investigate several versions of a cardinal characteristic $ \frak f$ defined by Frankiewicz. Vojtáš showed ${\frak b} \leq{\frak f}$ , and Blass showed ${\frak f} \leq \min({\frak d},{\mbox{\rm unif}}({\bf K}))$ . We show that all the versions coincide and that ${\frak f}$ is greater than or equal to the splitting number. We prove the consistency of $\max({\frak b},{\frak s}) <{\frak f}$ and of ${\frak f} < \min({\frak d},{\mbox{\rm unif}}({\bf K}))$.
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  24.  20
    Non-constructive galois-tukey connections.Heike Mildenberger - 1997 - Journal of Symbolic Logic 62 (4):1179-1186.
    There are inequalities between cardinal characteristics of the continuum that are true in any model of ZFC, but without a Borel morphism proving the inequality. We answer some questions from Blass [1].
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  25.  38
    On the homogeneity property for certain quantifier logics.Heike Mildenberger - 1992 - Archive for Mathematical Logic 31 (6):445-455.
    The local homogeneity property is defined as in [Mak]. We show thatL ωω(Q1) and some related logics do not have the local homogeneity property, whereas cofinality logicL ωω(Q cfω) has the homogeneity property. Both proofs use forcing and absoluteness arguments.
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  26.  61
    Order types of free subsets.Heike Mildenberger - 1997 - Annals of Pure and Applied Logic 89 (1):75-83.
    We give for ordinals α a lower bound for the least ordinal α such that Frordξ,β) and show that given enough measurable cardinals there are forcing extensions where the given bounds are sharp.
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  27.  3
    Specializing Aronszajn Trees with Strong Axiom A and Halving.Heike Mildenberger & Saharon Shelah - 2019 - Notre Dame Journal of Formal Logic 60 (4):587-616.
    We construct creature forcings with strong Axiom A that specialize a given Aronszajn tree. We work with tree creature forcing. The creatures that live on the Aronszajn tree are normed and have the halving property. We show that our models fulfill ℵ1=d
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  28.  24
    The club principle and the distributivity number.Heike Mildenberger - 2011 - Journal of Symbolic Logic 76 (1):34 - 46.
    We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with h > N₁ is consistent. We work with a class of axiom A forcings with countable conditions such that q ≥ n p is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for (...)
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  29.  14
    The minimal cofinality of an ultrapower of ω and the cofinality of the symmetric group can be larger than b+.Heike Mildenberger & Saharon Shelah - 2011 - Journal of Symbolic Logic 76 (4):1322-1340.
  30.  6
    The Strengths of Some Violations of Covering.Heike Mildenberger - 2001 - Mathematical Logic Quarterly 47 (3):291-298.
    We consider two models V1, V2 of ZFC such that V1 ⊆ V2, the cofinality functions of V1 and of V2 coincide, V1 and V2 have that same hereditarily countable sets, and there is some uncountable set in V2 that is not covered by any set in V1 of the same cardinality. We show that under these assumptions there is an inner model of V2 with a measurable cardinal κ of Mitchell order κ++. This technical result allows us to show (...)
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  31.  5
    Review: Andreas Blass, Haim Judah, Simple Cardinal Characteristics of the Continuum. [REVIEW]Heike Mildenberger - 2002 - Bulletin of Symbolic Logic 8 (4):552-553.
  32.  23
    Blass Andreas. Simple cardinal characteristics of the continuum. Set theory of the reals, edited by Judah Haim, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 63–90. [REVIEW]Heike Mildenberger - 2002 - Bulletin of Symbolic Logic 8 (4):552-553.